Before you even begin reading this build, it's important to realize that I'm not intending to make a tried and true, ultimate, competitive build. I'm a pretty casual LoL player, as far as League players go. Anivia is a champ that is very commonly played in ranked, but almost never in unranked. Although she can easily carry late game, Anivia isn't really considered to be OP'd by anyone's standards. If you're doing well with Anivia, you probably deserve it. I build Anivia pretty differently than most players, mainly for the fact that I build heavily in the Utility tree, as opposed to the Offense tree (I also didn't use to have any magic pen :P). I appreciate constructive criticism, however, don't waste your time commenting if all you're going to do is complain about how I'm not following the meta, or something equally pointless
Anivia's main stat is AP, as all of her skills scale with it. Also, Anivia is extremely vulnerable to ganks, especially at lower levels when you don't have completed boots, and higher leveled skills to slow/stun/block your attackers. Anivia is both one of, if not the, squishiest and slowest champ in the game. Armor is important in general, especially for Anivia, who has low armor, and can easily be crushed by AD-heavy champs if not careful. The extra armor'll help you survive longer. The AP, again, is self-explanatory.
I'm not a fan of Greater Glyph of Scaling Ability Power on Anivia, but if you're fond of them, you could have any combination of those and Greater Glyph of Ability Power.
Greater Mark of Magic Penetration: I used to use Greater Mark of Armor, since they give you about as much armor as these marks do magic pen, but that's building Anivia a bit too tanky for her role, so I switched my marks in favor of a higher damage output. I also used to use Boots of Swiftness as opposed to Sorcerer's Shoes. I've played Anivia in numerous ways in the past, so trust me when I say that magic pen is almost mandatory for her.
Greater Seal of Armor: These're probably the best seals in the game, and armor's always nice, especially since Anivia has very low armor. Since there aren't many other seals worth using, (mana regen runes're jokes) much less for Anivia, I suggest taking the tremendous armor boost.
Greater Glyph of Ability Power: I really believe that offensive runes're for the early game, and these give you some more damage. If you prefer, you could go for Greater Glyph of Force, but as I said, I prefer the early damage. Late game damage is gonna come from items.
Greater Glyph of Scaling Magic Resist: These're also useful runes, even for Anivia, but since I'm gonna start getting Athene's Unholy Grail (previously a worthless item for Anivia in 3s, imo), MR won't be much of a problem.
Greater Quintessence of Ability Power: These runes're much better than Greater Quintessence of Scaling Ability Power (on Anivia). These give you AP. AP increases your damage. Damage makes you hurt. You wanna hurt as this champ. Simple.
Greater Quintessence of Movement Speed: When using Sorcerer's Shoes, Anivia is simply too slow to secure most kills, or escape from lethal combat. Using these quints makes those much easier while still getting the magic pen from the boots.
I build heavily in the Utility tree, as opposed to the Offense tree. The simple reason for this is that the VAST majority of your AP/damage is gonna come from items anyway, and the extra damage granted from the Offense tree isn't enough of a lure for me. The lack of damage from the Offense tree is easily filled by items, while many of the bonuses granted from the Utility tree simply can't be filled with items that suit Anivia.
I've tried numerous mastery page set-ups for Anivia, from 0/9/21 and 0/8/22, to 21/0/9, and this is my current mastery page.
Sorcery : Combined with Intelligence , your passive CDR'll be 10%, which is solid.
Blast : This gets your AP rolling without even buying items, so you can squeeze Catalyst the Protector in right off the bat without sacrificing damage. Also, you need this in order to get...
Arcane Knowledge : Magic pen is a glorious thing on Anivia, and this gives you some more initial magic pen.
Summoner's Insight : I would take this mastery just for improved Clarity, and since I now use Flash (I used to use Ghost instead) too, it's a no brainer.
Meditation : Anivia is a very mana hungry champ. This gives you extra mana regen, and you still need more points to get deeper in the tree. Get these masteries to help with some extra mana.
Mastermind : This'll help you get Flash more often, as well as getting you deeper in the tree.
Expanded Mind : Anivia is a very mana hungry champ, and this gives you extra mana. Again, how in-depth do you need me to go? I'm assuming that you aren't that inept.
Greed : I know that this isn't the best mastery, but it'll give you some extra gold, and it's better for Anivia (keep in mind this is a TT guide) than any of the other masteries at this tier. Runic Affinity isn't that useful on this map since only the speed boost in the middle of the map and Vilemaw's buff would work with this. Spell vamp doesn't provide much for Anivia (especially a mere 3%), as most of her damage comes from AoE/multi-target spells, so Transmutation isn't very alluring.
Awareness : You're finally getting into the bottom of the tree. This mastery's really gonna help you get the level advantage over your opponents. That makes things easier. Yet again, most of these masteries're self-explanatory.
Strength of Spirit : While not nearly as useful as it used to be, this mastery is great for squishy mages who use mana. Since it isn't as good, I won't get nearly as in-depth as I did before, but this is worth investing one point in.
Intelligence : CDR isn't extremely useful on Anivia until you get a lot of it, but this helps in the early game, and late game.
Nimble : Extra MS on the extremely slow Anivia is great! This, along with the new boot upgrades, makes Anivia actually pretty fast, and fast is good.
The main portion of the build. Items are the single biggest game changers. If anything is confusing as to why I get the items I do, feel free to ask, however, I believe that the items're fairly self-explanatory.
If you're doing well, finish off with Void Staff.
If you aren't doing well, get Rylai's Crystal Scepter.
If you're facing two or more AP champs, and they're really hurting you, grab Abyssal Scepter.
You could do the extremely stereotypical Flash/ Ignite combo, however, I think that taking Clarity makes laning a lot easier. Since you'd only need Ignite for the grevious wound on Anivia, I don't think it's necessary. Ghost is one of the most underrated Summoner Spells, and, in fact, I used to use it in place of Flash for a long time. The only reasons that I starting using Flash instead is so that I could jump my own Crystallize, and while it isn't as stable as Ghost, it has the added usefulness of jumping over terrain. Also, if you're pretty sure that you're gonna die, this could be used to better position yourself for Rebirth, such as within a tower's range.